View source for Stein–Strömberg theorem

In [[mathematics]], the '''Stein–Strömberg theorem''' or '''Stein–Strömberg inequality''' is a result in [[measure theory]] concerning the [[Hardy–Littlewood maximal operator]].  The result is foundational in the study of the problem of [[differentiation of integrals]].  The result is named after the [[mathematician]]s [[Elias M. Stein]] and [[Jan-Olov Strömberg]].

==Statement of the theorem==
Let ''&lambda;''<sup>''n''</sup> denote ''n''-[[dimension]]al [[Lebesgue measure]] on ''n''-dimensional [[Euclidean space]] '''R'''<sup>''n''</sup> and let ''M'' denote the Hardy–Littlewood maximal operator: for a function ''f''&nbsp;:&nbsp;'''R'''<sup>''n''</sup>&nbsp;→&nbsp;'''R''', ''Mf''&nbsp;:&nbsp;'''R'''<sup>''n''</sup>&nbsp;→&nbsp;'''R''' is defined by

:<math>Mf(x) = \sup_{r > 0} \frac1{\lambda^{n} \big( B_{r} (x) \big)} \int_{B_{r} (x)} | f(y) | \, \mathrm{d} \lambda^{n} (y),</math>

where ''B''<sub>''r''</sub>(''x'') denotes the [[open ball]] of [[radius]] ''r'' with center ''x''.  Then, for each ''p''&nbsp;&gt;&nbsp;1, there is a constant ''C''<sub>''p''</sub>&nbsp;&gt;&nbsp;0 such that, for all [[natural number]]s ''n'' and functions ''f''&nbsp;∈&nbsp;''L''<sup>''p''</sup>('''R'''<sup>''n''</sup>;&nbsp;'''R'''),

:<math>\| Mf \|_{L^{p}} \leq C_{p} \| f \|_{L^{p}}.</math>

In general, a maximal operator ''M'' is said to be of '''strong type''' (''p'',&nbsp;''p'') if

:<math>\| Mf \|_{L^{p}} \leq C_{p, n} \| f \|_{L^{p}}</math>

for all ''f''&nbsp;∈&nbsp;''L''<sup>''p''</sup>('''R'''<sup>''n''</sup>;&nbsp;'''R''').  Thus, the Stein–Strömberg theorem is the statement that the Hardy–Littlewood maximal operator is of strong type (''p'',&nbsp;''p'') uniformly with respect to the dimension ''n''.

==References==
* {{cite journal
| last = Stein
| first = Elias M.
| authorlink = Elias M. Stein
|author2=Strömberg, Jan-Olov
| title = Behavior of maximal functions in '''R'''<sup>''n''</sup> for large ''n''
| journal = Ark. Mat.
| volume = 21
| year = 1983
| issue = 2
| pages = 259–269
| doi = 10.1007/BF02384314
| doi-access = free
}} {{MathSciNet|id=727348}}
* {{cite journal
| last = Tišer
| first = Jaroslav
| title = Differentiation theorem for Gaussian measures on Hilbert space
| journal = Trans. Amer. Math. Soc.
| volume = 308
| year = 1988
| issue = 2
| pages = 655&ndash;666
| doi = 10.2307/2001096
| jstor = 2001096
}} {{MathSciNet|id=951621}}

{{DEFAULTSORT:Stein-Stromberg theorem}}
[[Category:Inequalities]]
[[Category:Theorems in measure theory]]
[[Category:Operator theory]]